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Zero-temperature limit of one-dimensional Gibbs states via renormalization: the case of locally constant potentials

Published online by Cambridge University Press:  04 June 2010

J.-R. CHAZOTTES ,
J.-M. GAMBAUDO  and
E. UGALDE
J.-R. CHAZOTTES
Affiliation:
Centre de Physique Théorique, École Polytechnique, 91128 Palaiseau Cedex, France (email: [email protected])
J.-M. GAMBAUDO
Affiliation:
Laboratoire J. A. Dieudonné, UMR CNRS 6621, Université de Nice-Sophia Antipolis, 06108 Nice Cedex 2, France
E. UGALDE
Affiliation:
Instituto de Física, Universidad Autónoma de San Luis Potosí, San Luis Potosí SLP 78290, México
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Abstract

Let A be a finite set and let ϕ:Aℤ→ℝ be a locally constant potential. For each β>0 (‘inverse temperature’), there is a unique Gibbs measure μβϕ. We prove that as β→+∞, the family (μβϕ)β>0 converges (in the weak-* topology) to a measure that we characterize. This measure is concentrated on a certain subshift of finite type, which is a finite union of transitive subshifts of finite type. The two main tools are an approximation by periodic orbits and the Perron–Frobenius theorem for matrices à la Birkhoff. The crucial idea we bring is a ‘renormalization’ procedure which explains convergence and provides a recursive algorithm for computing the weights of the ergodic decomposition of the limit.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 4 , August 2011 , pp. 1109 - 1161
Copyright
Copyright © Cambridge University Press 2010

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References

[1]Birkhoff, G.. Projective contraction of a cone equipped with its associated Hilbert metric. Trans. Amer. Math. Soc. 85 (1957), 219–227.Google Scholar
[2]Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, 2nd edn.(Lecture Notes in Mathematics, 470). Springer, Berlin, 2008.CrossRefGoogle Scholar
[3]Brémont, J.. Gibbs measures at temperature zero. Nonlinearity 16 (2003), 419–426.CrossRefGoogle Scholar
[4]Chazottes, J.-R. and Hochman, M.. On the zero-temperature limit of Gibbs states. Comm. Math. Phys. (2010), to appear, http://www.springerlink.com/index/1083734146562t7g.pdf.Google Scholar
[5]Chazottes, J.-R., Ramírez, L. A. and Ugalde, E.. Finite type approximations to Gibbs measures on sofic subshifts. Nonlinearity 18(1) (2005), 445–465.Google Scholar
[6]Conze, J.-P. and Guivarc’h, Y.. Croissance des sommes ergodiques et principe variationnel. Unpublished manuscript, 1995.Google Scholar
[7]van Enter, A. C. D. and Ruszel, W. M.. Chaotic temperature dependence at zero temperature. J. Stat. Phys. 127 (2007), 567–573.CrossRefGoogle Scholar
[8]Georgii, H.-O.. Gibbs Measures and Phase Transitions (de Gruyter Studies in Mathematics, 9). Walter de Gruyter & Co., Berlin, 1988.CrossRefGoogle Scholar
[9]Jenkinson, O.. Geometric barycentres of invariant measures for circle maps. Ergod. Th. & Dynam. Sys. 21(2) (2001), 511–532.CrossRefGoogle Scholar
[10]Leplaideur, R.. A dynamical proof for the convergence of Gibbs measures at temperature zero. Nonlinearity 18 (2005), 2847–2880.CrossRefGoogle Scholar
[11]Nekhoroshev, N. N.. Asymptotics of Gibbs measures in one-dimensional lattice models. Moscow Univ. Math. Bull. 59(1) (2004), 10–15.Google Scholar
[12]Ruelle, D.. Thermodynamic Formalism. The Mathematical Structures of Equilibrium Statistical Mechanics, 2nd edn.(Cambridge Mathematical Library). Cambridge University Press, Cambridge, 2004.CrossRefGoogle Scholar
[13]Seneta, E.. Non-negative Matrices and Markov Chains (Springer Series in Statistics). Springer, Berlin, 1981.CrossRefGoogle Scholar
[14]van Enter, A. C. D., Fernández, R. and Sokal, A.. Regularity properties and pathologies of position-space renormalization-group transformations: scope and limitations of Gibbsian theory. J. Stat. Phys. 72(5–6) (1993), 879–1165.CrossRefGoogle Scholar